Useful Formulae

[oneokie]

Greenhill formula for those of us that are math challenged
For determining optimum twist rate for boolits.

For velocities less than 1800 fps

150 x D²/BL

For velocities more than 1800 fps

180 x D²/BL

D=diameter of bullet in inches
BL=bullet length in inches


greenhill formula post#5
http://castboolits.gunloads.com/showthread.php?t=22699


RPM formula
post#6
http://castboolits.gunloads.com/showthread.php?t=25226

[JIMinPHX]

This is a shortened version of what I posted on the old MarlinTalk board when it was still in existance several years ago -

Twist Rates

On another thread, I was tipped off to an article on barrel twist in a gun magazine by Gunjunkie. Since then I have done some further research. The original article was in the January issue of American Rifleman on page 30. It credits a Sir George Greenhill with developing the formula T x L = 150 for estimating the proper relationship of bullet length to barrel twist.

This looks really simple until you find out that the L is the length of the slug in bullet diameters, not in inches & T is the twist of the barrel in bullet diameters, not inches. Since they didn’t take the time to convert the formula into inches, I will do it here so that we can all have numbers to look at that actually mean something to us. I’ll leave the algebra derivations out unless someone wants me to post them, but this is what it comes down to:

The proper Twist in inches = 150 x caliber diameter x caliber diameter / Length of bullet in inches
Or for a .224 diameter bullet that is 3/4 (.75) of an inch long
Twist in inches = (150) x (.224) x (.224) / (.75)
Twist = a shade over 10 (or 1:10)

Also
The proper bullet length in inches = 150 x caliber diameter x caliber diameter / Twist in inches
Or in the case of a .223 Remington with a 1:12 twist like a NEF Handi-rifle:
Best bullet length in inches = (150) x (.224) x (.224) / 12
Best length = .627 (5/8 inch)

Assuming that Sir George’s formula speaks the truth, in the case of a .224 diameter bullet, the following twists would prefer the following bullet lengths:
Twist Bullet Length
1:7 – 1.075 inch
1:9 - .836 inch
1:10 - .753 inch
1:12 - .627 inch
1:14 - .538 inch

I pulled some .223 ammo apart & measured the following bullet lengths:
SS-109 green tip penetrater core M855 ball - .900 inch long
55gr FMJ M193 Ball - .730 inch long
55gr soft point pulled from Ultramax reload - .690 inch
50 gr Hornady soft point - .655 inch

I did some scrounging around after this & came up with the following info from various sources:

* The Greenhill Formula is a simplified method for determining mathematically the amount of spin necessary to stabilize a bullet. It was worked out in 1879 by Sir Alfred George Greenhill who was a Professor of Mathematics at Woolwich and teaching the Advanced British Artillery Officers Class. It was considered satisfactory for bullets having a density of .392 lbs/cubic inch or greater. (Lead has a density of .409 lbs/ cubic inch, and copper has a density of from .318-.325 lbs/cubic inch, depending on the alloy)

* The actual formula is much more complicated It is Gyroscopic Stability (GS) = the spin rate (in radians per second, squared) times the polar moment of inertia, squared, divided by the pitching moment coefficient derivative per sine of the angle of attack times the transverse moment of inertia times the air density times the velocity squared. (My keyboard does not have all the correct symbols and that is why I wrote it out). For the bullet to be stable, GS > 1.0. This is actually a short version as the pitching moment coefficient component is a complicated calculation that derives the center of gravity and the center of reverse air pressure. The equation is basically calculating the linear difference between the center of gravity and the center of reverse air pressure on the nose of the bullet. The greater the difference, the greater the spin required to keep the bullet pointed nose forward. It used to take me about three days to calculate one new design by hand. My computer does it in about 20 seconds, now.
Warren Jensen

* The number 150 is a constant used by Greenhill and works well at velocities in the vicinity of 1500 fps or greater. At 2800 fps the constant can be changed to 180 with good results.
Note that it is bullet LENGTH, not weight that is important. Greenhill works well with all lead/lead-alloys commonly used for bullets.

* Except for the extremely exacting requirements of bench rest shooters, and very long range shooters, an "over-stabilized" bullet (rifling twist faster than the minimum required for good stabilization) will shoot as well as a theoretically critically stabilized one.
Barrel twist is usually chosen for the longest bullet that will be fired in that gun. Accordingly, the military has fairly recently changed their .223 barrels to a 7 1/2" twist from a slower one for the newer, longer and heavier bullet that has become standard issue (68 grains, I believe).

* the faster the twist, the quicker it will wear out.

* As temperature or velocity decreases a faster twist is needed to maintain the same level of stability. Colder and thus denser air has a more destabilizing affect than warmer air. A lower muzzle velocity results in a slower rotational speed of the bullet and thus less stability.

*Another catch is that Greenhill assumes that the bullet's specific gravity is 10.9 (a lead cored jacketed bullet). For other bullet construction such as a steel core you need to apply a fudge factor by determining the bullets specific gravity. The formula would be:
Twist = [Square Root (10.9 / specific gravity of the new bullet)] * twist derived for a lead core bullet

*****After all this is said & done, I have found that my .223 Handi-rifle with a 1:12 twist shoots best with 50gr & below slugs.

Also, If a twist rate is too quick, it can put excessive centripetal force on a slug & cause it to vaporize as it leaves the barrel. Very light cased .22 varmint slugs are known for vaporizing when they leave 1:7 twist military barrels at over a quarter million RPM.

[zipdog]

Now I'm really confused. All the reading I have done suggests that the 1/18" twist of my .38-55 is too slow to stabilize a "heavy" (300+ gr) bullet, which means a long bullet, since with a given diameter the only way to add weight is to add length. The Greenhill formula results in a length of 1.14" for this caliber, which all the shooters say is too long. What am I missing, other than the fact that the formula is an approximation? Of course, the velocity would be more in the range of 1300-1400 fps. Would this affect the results significantly?

Rusty

[JIMinPHX]

Hi Rusty,

150 should be the right constant for the velocity range that you are using. Above 2000 or 3000 fps you can usually get away with using something like 180 for the constant instead.

Correct me if I’m wrong, but I believe that a 38-55 Winchester uses a real .38” diameter boolit & not a .357” like a .38 special does. When I do the math for a 1:18 twist & a .38” diameter boolit, I get Max boolit length = 150 x .38 x .38 / 18 = 1.2”

There is a little room for fluff here sometimes. If your boolits are 1-1/4” or less in length then you should expect good results. I don’t know which boolit you are using, but I would guestimate that a 300-grain boolit in .38” diameter should come out to just under an inch & a quarter long unless you are using something pointy.

[JIMinPHX]

I just took a look in my Lyman book. They list max speed for a 269 grain boolit in a 38-55 as less than 1100fps. If a 300-grain boolit is even slower than that, you may need to adjust the coefficient down below 150 & your max boolit length would shrink. I'm not sure how far. The Lyman book used a 1:18 twist Stevens model 44-1/2 for their tests.